arXiv research

Let K be a an alternating prime knot in the 3-sphere. We investigate the category of flypes between reduced alternating diagrams for K. As a consequence, we show that any odd prime order action on K is isotopic through maps of pairs to a single flype. This implies that for any odd prime order action on K there is eithe…

New invariants derived from Seifert graphs help distinguish alternating links.

We prove a Kauffman-Murasugi-Thistlethwaite theorem for alternating links in thickened surfaces. It states that any reduced alternating diagram of a link in a thickened surface has minimal crossing number, and any two reduced alternating diagrams of the same link have the same writhe. This result is proved more general…

The paper proves a generalized Kauffman-Harary conjecture for prime determinant links.

Tait's flyping conjecture, stating that two reduced, alternating, prime link diagrams can be connected by a finite sequence of flypes, is extended to reduced, alternating, prime diagrams of 4-regular graphs in S^3. The proof of this version of the flyping conjecture is based on the fact that the equivalence classes wit…

It is well known that the minimum crossing number of an alternating link equals the number of crossings in any reduced alternating link diagram of the link. This remarkable result is an application of the Jones polynomial. In the case of the braid index of an alternating link, Murasugi had conjectured that the number o…

A rational link may be represented by any of the (infinitely) many link diagrams corresponding to various continued fraction expansions of the same rational number. The continued fraction expansion of the rational number in which all signs are the same is called a {\em nonalternating form} and the diagram corresponding…

New invariant for special alternating links based on graph Laplacian.

It is shown that there exist alternating non-Montesinos knots whose essential spanning surfaces with maximal and minimal boundary slopes are not realised by the checkerboard surfaces coming from a reduced alternating planar diagram.

Classifies involutions on alternating prime non-split links.

Bankwitz characterized an alternating diagram representing the trivial knot. A non-alternating diagram is called almost alternating if one crossing change makes the diagram alternating. We characterize an almost alternaing diagram representing the trivial knot. As a corollary we determine an unknotting number one alter…

Constructs links that are both quasi-alternating and almost alternating.

Let $D$ be a reduced alternating diagram of a non-split link $L$ and $\tilde{L}$ be the link whose diagram is obtained from $D$ by a crossing change. If $\tilde{L}$ is alternating, then $c(\tilde{L})\leq c(L)-2$. In this paper we explore when $c(\tilde{L})=c(L)-2$ holds and obtain a simple sufficient and necessary cond…

The study proves a theorem for alternating knots in handlebodies.

We establish a characterization of alternating links in terms of definite spanning surfaces. We apply it to obtain a new proof of Tait's conjecture that reduced alternating diagrams of the same link have the same crossing number and writhe. We also deduce a result of Banks and Hirasawa-Sakuma about Seifert surfaces for…

The Penrose-Kauffman polynomial connects knot theory to graph coloring.

New theorem bounds link volume using surface coefficients.

We prove that if $L$ is a non-trivial alternating link embedded (without crossings) in a closed surface $F\subset S^3$, then $F$ has a compressing disk whose boundary intersects $L$ in no more than two points. Moreover, whenever the surface is incompressible and $\partial$-incompressible in the link exterior, it can be…

We prove the Kauffman-Harary Conjecture, posed in 1999: given a reduced, alternating diagram D of a knot with prime determinant p, every non-trivial Fox p-coloring of D will assign different colors to different arcs.

We partially solve the conjecture by A.Shumakovitch about torsion in the Khovanov homology of prime, non-split links in S^3. We give a size restriction on the Khovanov homology of almost alternating links. We relate the Khovanov homology of the connected sum of a link diagram and the Hopf link with the Khovanov homolog…

In 1999, Kauffman-Harary conjectured that every non-trivial Fox $p$-coloring of a reduced, alternating knot diagram with prime determinant $p$ is heterogeneous. Ten years later this conjecture was proved by W. Mattman and P. Solis. Mathew Williamson generalized this conjecture to alternating virtual knots and proved it…

A link is almost alternating if it is non-alternating and has a diagram that can be transformed into an alternating diagram via one crossing change. We give formulas for the first two and last two potential coefficients of the Jones polynomial of an almost alternating link. Using these formulas, we show that the Jones …

For an oriented virtual link, L.H. Kauffman defined the f-polynomial (Jones polynomial). The supporting genus of a virtual link diagram is the minimal genus of a surface in which the diagram can be embedded. In this paper we show that the span of the f-polynomial of an alternating virtual link L is determined by the nu…

Given a reduced alternating diagram for a link, we obtain conditions that guarantee that the link complement has a complete hyperbolic structure, crossing arcs are the edges of an ideal geodesic triangulation, and every crossing arc is isotopic to a simple geodesic. The latter was conjectured by Sakuma and Weeks in 199…

In this paper, we show that a link which has a positive and almost alternating diagram is alternating, besides that a positive and non-alternating Montesinos link has an almost positive-alternating diagram.

It is well known that the braid index of a link equals the minimum number of Seifert circles among all link diagrams representing it. For a link with a reduced alternating diagram $D$, $s(D)$, the number of Seifert circles in $D$, equals the braid index $\textbf{b}(D)$ of $D$ if $D$ contains no {\em lone crossings} (a …

Proves alternating quasipositive links are positive and strongly quasipositive.

Minimal grid diagrams for 12-crossing prime knots identified.

Bounding twist number of surface links using polynomial coefficients.

This paper finds all prime alternating knots with minimal warping degree two.

By using non-positively curved cubings of prime alternating link exteriors, we prove that certain ideal triangulations of their complements, derived from reduced alternating diagrams, are non-degenerate, in the sense that none of the edges is homotopic relative its endpoints to a peripheral arc. This guarantees that th…

We give a sufficient condition for an almost alternating link diagram to represent a non-splittable link. The main theorem gives us a way to see if a given almost alternating link diagram represents a splittable link without increasing numbers of crossings of diagrams in the process. Moreover, we show that almost alter…

The paper explores when specific knot operations simplify diagrams.

The study proves large alternating Montesinos knots cannot have purely cosmetic surgeries.

We study the head and tail of the colored Jones polynomial while focusing mainly on alternating links. Various ways to compute the colored Jones polynomial for a given link give rise to combinatorial identities for those power series. We further show that the head and tail functions only depend on the reduced checkerbo…

The Jones polynomial can be expressed in terms of spanning trees of the graph obtained by checkerboard coloring a knot diagram. We show there exists a complex generated by these spanning trees whose homology is the reduced Khovanov homology. The spanning trees provide a filtration on the reduced Khovanov complex and a …

The d-invariant of an integral, positive definite lattice L records the minimal norm of a characteristic covector in each equivalence class mod 2L. We prove that the 2-isomorphism type of a connected graph is determined by the d-invariant of its lattice of integral cuts (or flows). As an application, we prove that a re…

Let $L$ be an oriented link with an alternating diagram $D$. It is known that $L$ is a fibered link if and only if the surface $R$ obtained by applying Seifert's algorithm to $D$ is a Hopf plumbing. Here, we call $R$ a Hopf plumbing if $R$ is obtained by successively plumbing finite number of Hopf bands to a disk. In t…

Functor decomposes Khovanov spectra for non-alternating diagrams.

Minimal grid diagrams found for 13-crossing prime knots.

We prove that the genus of the Turaev surface of a link diagram is determined by a graph whose vertices correspond to the boundary components of the maximal alternating regions of the link diagram. Furthermore, we use these graphs to classify link diagrams whose Turaev surface has genus one or two, and we prove that si…

New tool lassos connects virtual and surface link diagrams, changing primeness rules.

The paper studies right-angled links on higher genus surfaces.

This is the third paper in a series devoted to enumerating the prime alternating knots and links. This paper establishes a method for enumerating the prime alternating links. It is shown that one may choose any prime alternating link diagram of a given minimal crossing size and by applications of just two operators (T …

We describe a model of random links based on random 4-valent maps, which can be sampled due to the work of Schaeffer. We will look at the relationship between the combinatorial information in the diagram and the hyperbolic volume. Specifically, we show that for random alternating diagrams, the expected hyperbolic volum…

We find all Heegaard diagrams with the property "alternating" or "weakly alternating" on a genus two orientable closed surface. Using these diagrams we give infinitely many genus two 3--manifolds, each admits an automorphism whose non-wondering set consists of two Williams solenoids, one attractor and one repeller. The…

Study links in 3-manifolds, linking volume to polynomial coefficients.

New minimal link diagrams found, including torus links and homogeneous ones.